I wish to solve the following two questions:
(a) Give a one-element generating set for the idealI of $\mathbb Z$ given by I= (14,49).
I solved this problem as follows:
Observe that this ideal must have the form $\{ 18x+49y | x,y \in \mathbb Z \}$ Now since $\gcd(14,49)=7$, we know all multiples of $14$ and $49$ are multiples of $7$, therefore contained in $7 \mathbb Z$. Similarly, we can write: $$14x'+ 49 y'=7 $$ by the extended Euclidean algorithm, therefore we know that we can generate all of $ 7 \mathbb Z$ in this fashion. We conclude $I=(\{7\})$.
(b) Give a one-element generating set for the ideal I of $\mathbb Z[\sqrt 3]$ given by $I= (8 +\sqrt3,122).$
I do not know how to tackle this problem. Can anyone provide a hint? I do not see any similarities with (a)..
Observe that $122$ is a multiple of the other element: $4(8 + \sqrt 3)(8 - \sqrt{3})=122$. Therefore $122$ is a redundant element, we can leave it out.